OFFSET
1,2
COMMENTS
A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..302
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 8*A(x))^n = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 8*A(x))^(n-1) = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 8*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 8*A(x)*x^n)^n = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 8*A(x)*x^n)^(n+1) = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 8*A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + 8*x^2 + 67*x^3 + 583*x^4 + 5209*x^5 + 47341*x^6 + 435366*x^7 + 4039863*x^8 + 37756884*x^9 + 354968162*x^10 + 3353718911*x^11 + 31818650141*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 8*A(x))^n = 1 - 10*x + 10*x^4 - 10*x^9 + 10*x^16 - 10*x^25 + 10*x^36 - 10*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06689190492526765287210924306086051922855300119805422530...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 8*A)^n = 5*(Pi/2)^(1/4)/gamma(3/4) - 4 = 0.5678956907805841070...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892197774017068345453024031418945825808997896316975979...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 8*A)^n = 5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 4 = 0.98319301552408211105...
(V.3) Let A = A(-exp(-Pi)) = -0.03328815108533045197898037729675109506494860109014140530...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 8*A)^n = 5*Pi^(1/4)/gamma(3/4) - 4 = 1.432174056066540072876...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001839973350611618077357159042562240768956638628903670470...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 8*A)^n = 5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 4 = 1.0186744274386954552...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 8*Ser(A))^m ) - 1 - 10*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/8 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved